Move object along local axis with Python API
Turns out having constraints on object is relevant – cannot use matrix_world for applying transforms. Instead use matrix_basis, to get access to location, rotation and scale, before constraints and parenting are applied.

Following code will move object along it's Z axis:

from mathutils import Matrix

loc = Matrix.Translation((0.0, 0.0, 1.5))
ob.matrix_basis *= loc